3.484 \(\int \frac{a+b \log (c (d (e+f x)^p)^q)}{\sqrt{g+h x}} \, dx\)

Optimal. Leaf size=103 \[ \frac{2 \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}+\frac{4 b p q \sqrt{f g-e h} \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{\sqrt{f} h}-\frac{4 b p q \sqrt{g+h x}}{h} \]

[Out]

(-4*b*p*q*Sqrt[g + h*x])/h + (4*b*Sqrt[f*g - e*h]*p*q*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]])/(Sqrt[
f]*h) + (2*Sqrt[g + h*x]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/h

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Rubi [A]  time = 0.136962, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {2395, 50, 63, 208, 2445} \[ \frac{2 \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}+\frac{4 b p q \sqrt{f g-e h} \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{\sqrt{f} h}-\frac{4 b p q \sqrt{g+h x}}{h} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d*(e + f*x)^p)^q])/Sqrt[g + h*x],x]

[Out]

(-4*b*p*q*Sqrt[g + h*x])/h + (4*b*Sqrt[f*g - e*h]*p*q*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]])/(Sqrt[
f]*h) + (2*Sqrt[g + h*x]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/h

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 2445

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

Rubi steps

\begin{align*} \int \frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt{g+h x}} \, dx &=\operatorname{Subst}\left (\int \frac{a+b \log \left (c d^q (e+f x)^{p q}\right )}{\sqrt{g+h x}} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{2 \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\operatorname{Subst}\left (\frac{(2 b f p q) \int \frac{\sqrt{g+h x}}{e+f x} \, dx}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{4 b p q \sqrt{g+h x}}{h}+\frac{2 \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\operatorname{Subst}\left (\frac{(2 b (f g-e h) p q) \int \frac{1}{(e+f x) \sqrt{g+h x}} \, dx}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{4 b p q \sqrt{g+h x}}{h}+\frac{2 \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\operatorname{Subst}\left (\frac{(4 b (f g-e h) p q) \operatorname{Subst}\left (\int \frac{1}{e-\frac{f g}{h}+\frac{f x^2}{h}} \, dx,x,\sqrt{g+h x}\right )}{h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{4 b p q \sqrt{g+h x}}{h}+\frac{4 b \sqrt{f g-e h} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{\sqrt{f} h}+\frac{2 \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}\\ \end{align*}

Mathematica [A]  time = 0.289237, size = 89, normalized size = 0.86 \[ \frac{2 \left (\sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )-2 b p q\right )+\frac{2 b p q \sqrt{f g-e h} \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{\sqrt{f}}\right )}{h} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d*(e + f*x)^p)^q])/Sqrt[g + h*x],x]

[Out]

(2*((2*b*Sqrt[f*g - e*h]*p*q*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]])/Sqrt[f] + Sqrt[g + h*x]*(a - 2*
b*p*q + b*Log[c*(d*(e + f*x)^p)^q])))/h

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Maple [A]  time = 0.335, size = 155, normalized size = 1.5 \begin{align*} 2\,{\frac{\sqrt{hx+g}a}{h}}+2\,{\frac{b\sqrt{hx+g}}{h}\ln \left ( c \left ( d \left ({\frac{f \left ( hx+g \right ) +eh-fg}{h}} \right ) ^{p} \right ) ^{q} \right ) }-4\,{\frac{bqp\sqrt{hx+g}}{h}}+4\,{\frac{bqpe}{\sqrt{ \left ( eh-fg \right ) f}}\arctan \left ({\frac{f\sqrt{hx+g}}{\sqrt{ \left ( eh-fg \right ) f}}} \right ) }-4\,{\frac{bfgpq}{h\sqrt{ \left ( eh-fg \right ) f}}\arctan \left ({\frac{f\sqrt{hx+g}}{\sqrt{ \left ( eh-fg \right ) f}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(d*(f*x+e)^p)^q))/(h*x+g)^(1/2),x)

[Out]

2/h*(h*x+g)^(1/2)*a+2/h*b*ln(c*(d*((f*(h*x+g)+e*h-f*g)/h)^p)^q)*(h*x+g)^(1/2)-4*b*p*q*(h*x+g)^(1/2)/h+4*b*q*p/
((e*h-f*g)*f)^(1/2)*arctan(f*(h*x+g)^(1/2)/((e*h-f*g)*f)^(1/2))*e-4/h*b*q*p*f/((e*h-f*g)*f)^(1/2)*arctan(f*(h*
x+g)^(1/2)/((e*h-f*g)*f)^(1/2))*g

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))/(h*x+g)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.52207, size = 483, normalized size = 4.69 \begin{align*} \left [\frac{2 \,{\left (b p q \sqrt{\frac{f g - e h}{f}} \log \left (\frac{f h x + 2 \, f g - e h + 2 \, \sqrt{h x + g} f \sqrt{\frac{f g - e h}{f}}}{f x + e}\right ) +{\left (b p q \log \left (f x + e\right ) - 2 \, b p q + b q \log \left (d\right ) + b \log \left (c\right ) + a\right )} \sqrt{h x + g}\right )}}{h}, \frac{2 \,{\left (2 \, b p q \sqrt{-\frac{f g - e h}{f}} \arctan \left (-\frac{\sqrt{h x + g} f \sqrt{-\frac{f g - e h}{f}}}{f g - e h}\right ) +{\left (b p q \log \left (f x + e\right ) - 2 \, b p q + b q \log \left (d\right ) + b \log \left (c\right ) + a\right )} \sqrt{h x + g}\right )}}{h}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))/(h*x+g)^(1/2),x, algorithm="fricas")

[Out]

[2*(b*p*q*sqrt((f*g - e*h)/f)*log((f*h*x + 2*f*g - e*h + 2*sqrt(h*x + g)*f*sqrt((f*g - e*h)/f))/(f*x + e)) + (
b*p*q*log(f*x + e) - 2*b*p*q + b*q*log(d) + b*log(c) + a)*sqrt(h*x + g))/h, 2*(2*b*p*q*sqrt(-(f*g - e*h)/f)*ar
ctan(-sqrt(h*x + g)*f*sqrt(-(f*g - e*h)/f)/(f*g - e*h)) + (b*p*q*log(f*x + e) - 2*b*p*q + b*q*log(d) + b*log(c
) + a)*sqrt(h*x + g))/h]

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Sympy [A]  time = 25.1486, size = 347, normalized size = 3.37 \begin{align*} \begin{cases} - \frac{\frac{2 a g}{\sqrt{g + h x}} + 2 a \left (- \frac{g}{\sqrt{g + h x}} - \sqrt{g + h x}\right ) + 2 b g \left (\frac{2 f p q \operatorname{atan}{\left (\frac{1}{\sqrt{\frac{f}{e h - f g}} \sqrt{g + h x}} \right )}}{\sqrt{\frac{f}{e h - f g}} \left (e h - f g\right )} + \frac{\log{\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{\sqrt{g + h x}}\right ) + 2 b \left (- \frac{2 f p q \left (- \frac{h \sqrt{g + h x}}{f} - \frac{h \operatorname{atan}{\left (\frac{1}{\sqrt{\frac{f}{e h - f g}} \sqrt{g + h x}} \right )}}{f \sqrt{\frac{f}{e h - f g}}}\right )}{h} - g \left (\frac{2 f p q \operatorname{atan}{\left (\frac{1}{\sqrt{\frac{f}{e h - f g}} \sqrt{g + h x}} \right )}}{\sqrt{\frac{f}{e h - f g}} \left (e h - f g\right )} + \frac{\log{\left (c \left (d \left (e - \frac{f g}{h} + \frac{f \left (g + h x\right )}{h}\right )^{p}\right )^{q} \right )}}{\sqrt{g + h x}}\right ) - \sqrt{g + h x} \log{\left (c \left (d \left (e - \frac{f g}{h} + \frac{f \left (g + h x\right )}{h}\right )^{p}\right )^{q} \right )}\right )}{h} & \text{for}\: h \neq 0 \\\frac{a x + b \left (- f p q \left (- \frac{e \left (\begin{cases} \frac{x}{e} & \text{for}\: f = 0 \\\frac{\log{\left (e + f x \right )}}{f} & \text{otherwise} \end{cases}\right )}{f} + \frac{x}{f}\right ) + x \log{\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )}{\sqrt{g}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(d*(f*x+e)**p)**q))/(h*x+g)**(1/2),x)

[Out]

Piecewise((-(2*a*g/sqrt(g + h*x) + 2*a*(-g/sqrt(g + h*x) - sqrt(g + h*x)) + 2*b*g*(2*f*p*q*atan(1/(sqrt(f/(e*h
- f*g))*sqrt(g + h*x)))/(sqrt(f/(e*h - f*g))*(e*h - f*g)) + log(c*(d*(e + f*x)**p)**q)/sqrt(g + h*x)) + 2*b*(
-2*f*p*q*(-h*sqrt(g + h*x)/f - h*atan(1/(sqrt(f/(e*h - f*g))*sqrt(g + h*x)))/(f*sqrt(f/(e*h - f*g))))/h - g*(2
*f*p*q*atan(1/(sqrt(f/(e*h - f*g))*sqrt(g + h*x)))/(sqrt(f/(e*h - f*g))*(e*h - f*g)) + log(c*(d*(e - f*g/h + f
*(g + h*x)/h)**p)**q)/sqrt(g + h*x)) - sqrt(g + h*x)*log(c*(d*(e - f*g/h + f*(g + h*x)/h)**p)**q)))/h, Ne(h, 0
)), ((a*x + b*(-f*p*q*(-e*Piecewise((x/e, Eq(f, 0)), (log(e + f*x)/f, True))/f + x/f) + x*log(c*(d*(e + f*x)**
p)**q)))/sqrt(g), True))

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Giac [A]  time = 1.25908, size = 173, normalized size = 1.68 \begin{align*} -\frac{2 \,{\left ({\left (2 \, f{\left (\frac{{\left (f g - h e\right )} \arctan \left (\frac{\sqrt{h x + g} f}{\sqrt{-f^{2} g + f h e}}\right )}{\sqrt{-f^{2} g + f h e} f} + \frac{\sqrt{h x + g}}{f}\right )} - \sqrt{h x + g} \log \left (f x + e\right )\right )} b p q - \sqrt{h x + g} b q \log \left (d\right ) - \sqrt{h x + g} b \log \left (c\right ) - \sqrt{h x + g} a\right )}}{h} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))/(h*x+g)^(1/2),x, algorithm="giac")

[Out]

-2*((2*f*((f*g - h*e)*arctan(sqrt(h*x + g)*f/sqrt(-f^2*g + f*h*e))/(sqrt(-f^2*g + f*h*e)*f) + sqrt(h*x + g)/f)
- sqrt(h*x + g)*log(f*x + e))*b*p*q - sqrt(h*x + g)*b*q*log(d) - sqrt(h*x + g)*b*log(c) - sqrt(h*x + g)*a)/h